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A368924
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Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.
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12
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1, 0, 1, 0, 2, 1, 1, 9, 6, 1, 15, 68, 48, 12, 1, 222, 720, 510, 150, 20, 1, 3670, 9738, 6825, 2180, 360, 30, 1, 68820, 159628, 110334, 36960, 6895, 735, 42, 1, 1456875, 3067320, 2090760, 721560, 145530, 17976, 1344, 56, 1, 34506640, 67512798, 45422928, 15989232, 3402756, 463680, 40908, 2268, 72, 1
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OFFSET
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0,5
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COMMENTS
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The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
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LINKS
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FORMULA
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E.g.f.: A(x,y) = exp(-log(1-T(x))/2 - T(x)/2 + y*T(x) - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 14 2024
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EXAMPLE
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Triangle begins:
1
0 1
0 2 1
1 9 6 1
15 68 48 12 1
222 720 510 150 20 1
3670 9738 6825 2180 360 30 1
68820 159628 110334 36960 6895 735 42 1
Row n = 3 counts the following loop-graphs:
{{1,2},{1,3},{2,3}} {{1},{1,2},{1,3}} {{1},{2},{1,3}} {{1},{2},{3}}
{{1},{1,2},{2,3}} {{1},{2},{2,3}}
{{1},{1,3},{2,3}} {{1},{3},{1,2}}
{{2},{1,2},{1,3}} {{1},{3},{2,3}}
{{2},{1,2},{2,3}} {{2},{3},{1,2}}
{{2},{1,3},{2,3}} {{2},{3},{1,3}}
{{3},{1,2},{1,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}], {n}], Count[#, {_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]], {n, 0, 5}, {k, 0, n}]
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PROG
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(PARI) T(n)={my(t=-lambertw(-x + O(x*x^n))); [Vecrev(p) | p <- Vec(serlaplace(exp(-log(1-t)/2 - t/2 + t*y - t^2/4)))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 14 2024
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CROSSREFS
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Cf. A000169, A057500, A062740, A129271, A133686, A322661, A367869, A367902, A368601, A368835, A368836, A368927.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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